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Noise sensitivity of random walks on groups
Itaï Benjaminia b, Jérémie Brieussela c d
- aDepartment of Mathematics , Weizmann Institute of Science , Rehovot , Israel
- bDepartment of Computer Science and Applied Mathematics [Rehovot] (Faculty of Mathematics and Computer Science The Weizmann Institute of Science POB 26, Rehovot 76100, ISRAEL - Israel)
- cInstitut Montpellierain Alexander Grothendieck , Université de Montpellier , France
- dIMAG - Institut Montpelliérain Alexander Grothendieck (UMR CNRS 5149 - Université Montpellier 2, Case courrier 051, 34095 Montpellier cedex 5 - France - France)
Latin American Journal of Probability and Mathematical Statistics2023-01-01 5 citations36 references
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Abstract
A random walk on a group is noise sensitive if resampling every step independently with a small probability results in an almost independent output.We precisely define two notions: ℓ 1 -noise sensitivity and entropy noise sensitivity.Groups with one of these properties are necessarily Liouville.Homomorphisms to free abelian groups provide an obstruction to ℓ 1 -noise sensitivity.We also provide examples of ℓ 1 and entropy noise sensitive random walks.Noise sensitivity raises many open questions which are described at the end of the paper.